First post here, I hope it's useful to someone
Sorry I don't have pictures of the whole process. I was actually expecting that I'll fry the thing, so I can only provide images of the finished thing. Pictures at the bottom of the post.

Since non-replaceable internal batteries are the bane of my existence, I decided to try and see if I could fit the Sansa Clip+, which is an amazing music player, with replaceable Eneloop batteries. I have never once soldered before in my life (or actually modified any existing hardware), so in that regard, I'm a complete novice. That's why I'm confident that even a novice can pull this off unless they have two left hands.

Items needed:
- 3x AA(A) Battery, depending on how large you want to get
- Some adhesive tape
- Hot glue
- A battery case that puts your three AA(A) type batteries in series ( http://image.dhgate.com/albu_375623019_00/1.0x0.jpg ) usually you can get those cheap (I paid like 2 bucks) from Amazon or something.

Since I have built a magnetic stirrer just a week earlier and bought a stabilized power supply for that, I tested whether or not the voltages will fry the Sansa Clip+ before I actually went through the trouble of putting everything together.
3 freshly charged Eneloops can supply up to 1.5 Volts each (and disposable ones will, too). So that's 4.5 V on the circuit instead of the maximum 4.2 V of a Li-Poly battery. This concerned me a little, but since the difference wasn't too great, I thought it was worth a try. With my stabilized power supply, I confirmed that 4.5 V is no issue at all for the player.

The next concern was that the Eneloops could possibly be discharged too deeply and being damaged by that, so I checked when the Sansa Clip+ will shut down due to undervoltage. It turns off at about 2.8 V, which means 0.933 V per battery and is still slightly above the minimum voltage of 0.9 V. Since I don't know much about electronics and I couldn't figure out an easy circuit to shut down earlier, I just left it at that. Tests later showed that after a complete discharge (the player shut down), my batteries still had 1.13 - 1.14 V, which is no issue at all. So maybe the player checks the voltage only every X seconds and 2.8 V is the point at which the player simply can't operate any more.

Once you opened the Clip+, you'll see the battery connected to the Clip+ by three wires (Red, Blue and Black). You can decide to leave them attached or de-solder them while you work on it. The battery itself is glued to the chip beneath it. Use a small screwdriver and carefully remove the battery from the chip. Do not deform the battery, as that could short the battery internally, causing it to bulge and possibly catch fire. When you got it off the chip, you need to remove the tape that encases a small PCB. Once you did that, carefully de-solder the battery from the small PCB (just two contacts. Battery + and -). Now solder the battery case wires to the small PCB you just removed from the original battery. In case you have forgotten which wire goes where, it's printed on the PCB next to the solder joints. If you got a red and black wire from the battery case, red will be +, black will be -. Otherwise it should be visible on the case somewhere.

If the battery case is connected to the small PCB, re-solder the three small wires coming from it to the Clip+ PCB if you de-soldered them. Now cover the small PCB with some adhesive tape from both sides to keep it from touching anything that conducts electricity inside the Clip+ and make a little hole in the Clip's casing for the wires of the battery case to go through (being lazy, I used my soldering iron for that). To prevent any mechanical strain on the soldering joints, cover the cables with hot glue right next to the opening (I did this from the outside, but I suppose you could do it from the inside too).

Now you can glue the Clip+ to the battery case. You'll need to be quick about it, since the hot glue will harden rather fast. You can add some extra glue from the sides of the Clip to be sure and stop the two wires from the case from moving around too much. Insert batteries and be happy about your (albeit ugly) Sansa Clip+ with replaceable batteries. If you use a micro SD card in your Clip+, it won't start up after it's been disconnected from power entirely (when switching batteries). Just remove the micro SD, turn it on and re-insert it. As long as it's not disconnected from power again, it'll start up without issues. The nice side effect is that you'll have increased runtime now. The Clip+ will display the remaining power incorrectly though. It will drop down to 50-60% fast, stay there for a long while and then slowly drop further (this is due to the chemistry of the batteries). The last 15% or so will drop quickly again.

The player may be larger now, but it also offers quick battery replacement (takes maybe 20 seconds) and longer runtimes. In theory you could also charge the batteries through the player, but they won't be charged fully and it will take a very long time.
For Eneloops I recommend either the BC-700 charger or the smart Sanyo/Panasonic charger.